I am trying to prove this and found the proof, but have no idea how we are able to multiply by (B-1A-1) in the first step when it is not in the original statement. I understand it, but not where this comes from. Also, on the second side, where does the AB come from?
Question: Prove: (AB)-1 = B-1A-1.
Solution: Using the associativity of matrix multiplication,
(AB)(B-1A-1) = A(BB-1)A-1 = AIA-1 = AA-1 = I
(B-1A-1)(AB) = B(AA-1)B-1 = BIB-1 = BB-1 = I:
Thus AB is invertible and B-1A-1 is its inverse